Lagrange polynomial

Lagrange polynomial
In numerical analysis, the Lagrange interpolating polynomial is the unique polynomial of lowest degree that interpolates a given set of data.
Given a data set of coordinate pairs with the are called nodes and the are called values. The Lagrange polynomial has degree and assumes each value at the corresponding node,
Although named after Joseph-Louis Lagrange, who published it in 1795, the method was first discovered in 1779 by Edward Waring. It is also an easy consequence of a formula published in 1783 by Leonhard Euler.
Uses of Lagrange polynomials include the Newton–Cotes method of numerical integration, Shamir's secret sharing scheme in cryptography, and Reed–Solomon error correction in coding theory.
For equispaced nodes, Lagrange interpolation is susceptible to Runge's phenomenon of large oscillation.
Definition
Given a set of nodes , which must all be distinct, for indices , the Lagrange basis for polynomials of degree for those nodes is the set of polynomials each of degree which take values if and . Using the Kronecker delta this can be written Each basis polynomial can be explicitly described by the product:
Notice that the numerator has roots at the nodes while the denominator scales the resulting polynomial so that
The Lagrange interpolating polynomial for those nodes through the corresponding values is the linear combination:
Each basis polynomial has degree , so the sum has degree , and it interpolates the data because
The interpolating polynomial is unique. Proof: assume the polynomial of degree interpolates the data. Then the difference is zero at distinct nodes But the only polynomial of degree with more than roots is the constant zero function, so or
Barycentric form
Each Lagrange basis polynomial can be rewritten as the product of three parts, a function common to every basis polynomial, a node-specific constant (called the barycentric weight), and a part representing the displacement from to :
By factoring out from the sum, we can write the Lagrange polynomial in the so-called first barycentric form:
If the weights have been pre-computed, this requires only operations compared to for evaluating each Lagrange basis polynomial individually.
The barycentric interpolation formula can also easily be updated to incorporate a new node by dividing each of the , by and constructing the new as above.
For any because the constant function is the unique polynomial of degree interpolating the data We can thus further simplify the barycentric formula by dividing
This is called the second form or true form of the barycentric interpolation formula.
This second form has advantages in computation cost and accuracy: it avoids evaluation of ; the work to compute each term in the denominator has already been done in computing and so computing the sum in the denominator costs only addition operations; for evaluation points which are close to one of the nodes , catastrophic cancelation would ordinarily be a problem for the value , however this quantity appears in both numerator and denominator and the two cancel leaving good relative accuracy in the final result.
Using this formula to evaluate at one of the nodes will result in the indeterminate ; computer implementations must replace such results by
Each Lagrange basis polynomial can also be written in barycentric form:
A perspective from linear algebra
Solving an interpolation problem leads to a problem in linear algebra amounting to inversion of a matrix. Using a standard monomial basis for our interpolation polynomial , we must invert the Vandermonde matrix to solve for the coefficients of . By choosing a better basis, the Lagrange basis we merely get the identity matrix which is its own inverse: the Lagrange basis automatically inverts the analog of the Vandermonde matrix.
This construction is analogous to the Chinese remainder theorem. Instead of checking for remainders of integers modulo prime numbers, we are checking for remainders of polynomials when divided by linears.
Furthermore, when the order is large, Fast Fourier transformation can be used to solve for the coefficients of the interpolated polynomial.
Example
We wish to interpolate over the domain at the three nodes
The node polynomial is
The barycentric weights are
The Lagrange basis polynomials are
The Lagrange interpolating polynomial is:
In (second) barycentric form,
Notes
The Lagrange form of the interpolation polynomial shows the linear character of polynomial interpolation and the uniqueness of the interpolation polynomial. Therefore, it is preferred in proofs and theoretical arguments. Uniqueness can also be seen from the invertibility of the Vandermonde matrix, due to the non-vanishing of the Vandermonde determinant.
But, as can be seen from the construction, each time a node xk changes, all Lagrange basis polynomials have to be recalculated. A better form of the interpolation polynomial for practical (or computational) purposes is the barycentric form of the Lagrange interpolation (see below) or Newton polynomials.
Lagrange and other interpolation at equally spaced points, as in the example above, yield a polynomial oscillating above and below the true function. This behaviour tends to grow with the number of points, leading to a divergence known as Runge's phenomenon; the problem may be eliminated by choosing interpolation points at Chebyshev nodes.
The Lagrange basis polynomials can be used in numerical integration to derive the Newton–Cotes formulas.
Remainder in Lagrange interpolation formula
When interpolating a given function f by a polynomial of degree at the nodes we get the remainder which can be expressed as
where is the notation for divided differences. Alternatively, the remainder can be expressed as a contour integral in complex domain as
The remainder can be bound as
Derivation
Clearly, is zero at nodes. To find at a point , define a new function and choose where is the constant we are required to determine for a given . We choose so that has zeroes (at all nodes and ) between and (including endpoints). Assuming that is -times differentiable, since and are polynomials, and therefore, are infinitely differentiable, will be -times differentiable. By Rolle's theorem, has zeroes, has zeroes... has 1 zero, say . Explicitly writing :
(Because the highest power of in is )
The equation can be rearranged as
Since we have
Derivatives
The th derivative of a Lagrange interpolating polynomial can be written in terms of the derivatives of the basis polynomials,
Recall (see above) that each Lagrange basis polynomial is
The first derivative can be found using the product rule:
The second derivative is
The third derivative is
and likewise for higher derivatives.
Finite fields
The Lagrange polynomial can also be computed in finite fields. This has applications in cryptography, such as in Shamir's Secret Sharing scheme.
See also
Neville's algorithm
Newton form of the interpolation polynomial
Bernstein polynomial
Carlson's theorem
Lebesgue constant (interpolation)
The Chebfun system
Table of Newtonian series
Frobenius covariant
Sylvester's formula
Finite difference coefficient
Hermite interpolation
References
External links
ALGLIB has an implementations in C++ / C# / VBA / Pascal.
GSL has a polynomial interpolation code in C
SO has a MATLAB example that demonstrates the algorithm and recreates the first image in this article
Lagrange Method of Interpolation — Notes, PPT, Mathcad, Mathematica, MATLAB, Maple at Holistic Numerical Methods Institute
Lagrange interpolation polynomial on www.math-linux.com
Dynamic Lagrange interpolation with JSXGraph
Numerical computing with functions: The Chebfun Project
Excel Worksheet Function for Bicubic Lagrange Interpolation
Lagrange polynomials in Python
Interpolation
Polynomials
Articles containing proofs